Four kinds of mathematics
When it comes to solving famous mathematical problems, there are four kinds of papers.
FIRST KIND. "This problem has been around for 50 years, and in this paper we present a complete solution." The paper gets checked by experts, and if it's correct, gets published in Annals of Mathematics, the author gets all the accolade, including $$$. Happy end.
SECOND KIND. "This problem has been around for 50 years, and in this paper we present a partial solution." The paper gets checked by experts, and if it's correct, gets published in Advances in Mathematics, the author gets some accolade and may get promoted if he/she manages to monetise their result.
THIRD KIND. "This problem has been around for 50 years, and we have no idea how to make any inroads, so we consider a similar problem and solve it." Everybody yawns, and some unfortunate PhD student is assigned the pitiful task to check it, which might end his/her career in mathematics. Good example: the Littlewood conjecture.
FOURTH KIND. "This problem has been around for 50 years, and we have no idea how to make any inroads. Moreover, we don't have any variations in mind, so we prove a bunch of technical results which have something to do with the original problem." This paper is read by no one and gets published in some third-rate journal, where the referee just rubber-stamps it.
What's the moral? Well, first kind is for the chosen few. If you are not one of them, make up your own shit, develop theories, which a dozen people on Earth can understand. This approach is monetisable, so you will make a good career, don't worry.
FIRST KIND. "This problem has been around for 50 years, and in this paper we present a complete solution." The paper gets checked by experts, and if it's correct, gets published in Annals of Mathematics, the author gets all the accolade, including $$$. Happy end.
SECOND KIND. "This problem has been around for 50 years, and in this paper we present a partial solution." The paper gets checked by experts, and if it's correct, gets published in Advances in Mathematics, the author gets some accolade and may get promoted if he/she manages to monetise their result.
THIRD KIND. "This problem has been around for 50 years, and we have no idea how to make any inroads, so we consider a similar problem and solve it." Everybody yawns, and some unfortunate PhD student is assigned the pitiful task to check it, which might end his/her career in mathematics. Good example: the Littlewood conjecture.
FOURTH KIND. "This problem has been around for 50 years, and we have no idea how to make any inroads. Moreover, we don't have any variations in mind, so we prove a bunch of technical results which have something to do with the original problem." This paper is read by no one and gets published in some third-rate journal, where the referee just rubber-stamps it.
What's the moral? Well, first kind is for the chosen few. If you are not one of them, make up your own shit, develop theories, which a dozen people on Earth can understand. This approach is monetisable, so you will make a good career, don't worry.